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Theorem elprg 3791
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )

Proof of Theorem elprg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2410 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 eqeq1 2410 . . 3  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
31, 2orbi12d 691 . 2  |-  ( x  =  A  ->  (
( x  =  B  \/  x  =  C )  <->  ( A  =  B  \/  A  =  C ) ) )
4 dfpr2 3790 . 2  |-  { B ,  C }  =  {
x  |  ( x  =  B  \/  x  =  C ) }
53, 4elab2g 3044 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721   {cpr 3775
This theorem is referenced by:  elpr  3792  elpr2  3793  elpri  3794  eltpg  3811  ifpr  3816  prid1g  3870  ordunpr  4765  hashtpg  11646  cnsubrg  16714  atandm  20669  nbgra0nb  21394  eupath2lem1  21652  eliccioo  24130  eldifpr  24345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781
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